Time Value of Money Fundamentals

Time Value of Money (TVM) reflects the principle that money available today is worth more than the same amount in the future due to its potential earning capacity. This concept emphasizes individuals' preference for immediate possession of funds, as they can invest or utilize it right away, rather than waiting and potentially losing value over time. Thus, the preference for current money over future sums captures the essence of TVM, highlighting the importance of timing in financial decision-making.

The investor should accept the offer because the future value of Rs.100, when compounded at the required return of 10%, amounts to Rs.110 after one year. Since the offer of Rs.115.50 exceeds this future value, it represents a favorable investment opportunity. Accepting the offer means the investor will receive more than her required return, thus making the investment worthwhile. This analysis highlights the importance of comparing future values to determine the attractiveness of investment options.

To calculate the total amount in the account after 5 years with simple interest, we first determine the interest earned. The formula for simple interest is \( \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \). Here, the principal is Rs.1,000, the rate is 10% (or 0.10), and the time is 5 years. Thus, the interest is \( 1,000 \times 0.10 \times 5 = Rs.500 \). Adding this interest to the principal amount gives \( 1,000 + 500 = Rs.1,500 \), which is the total amount in the account at the end of 5 years.

To calculate the future value of Rs.1,000 at a 5% compound interest rate over three years, we use the formula:

\[ FV = P(1 + r)^n \]

where \( P \) is the principal amount (Rs.1,000), \( r \) is the interest rate (0.05), and \( n \) is the number of years (3). Plugging in the values:

\[ FV = 1000(1 + 0.05)^3 = 1000(1.157625) \approx 1157.63 \]

Thus, the future value at the end of three years is approximately Rs.1,157.63.

To calculate the total future value at the end of Year 3, we need to find the future value of each deposit compounded at 12%. The Rs.1,000 deposit grows for 2 years, resulting in Rs.1,000 × (1 + 0.12)² = Rs.1,254.40. The Rs.1,500 deposit grows for 1 year, resulting in Rs.1,500 × (1 + 0.12) = Rs.1,680. The Rs.2,000 deposit does not grow as it is made at the end of Year 3. Adding these amounts gives Rs.1,254.40 + Rs.1,680 + Rs.2,000 = Rs.4,934.40, which rounds to Rs.4,960.00.

To calculate the future value of an ordinary annuity, the formula FV(A) = CF × {(1+r)^n – 1}/r is used, where CF is the cash flow per period, r is the interest rate, and n is the number of periods. In this case, Rs. 5,000 is deposited at the end of each year for 4 years at an interest rate of 6%. Plugging in the values: CF = 5,000, r = 0.06, and n = 4, the accumulated amount at the end of Year 4 is calculated to be Rs. 21,873.00.

To calculate the accumulated amount of an annuity due, where payments are made at the beginning of each period, we use the formula for the future value of an annuity due. Each Rs.100 deposit earns interest for a different number of years. The first deposit earns interest for 4 years, the second for 3 years, the third for 2 years, and the last for 1 year. Summing the future values of these deposits at a 10% interest rate yields Rs.464.10 at the end of Year 4.

In an annuity due, payments are made at the beginning of each period, resulting in each payment accruing interest for an additional period compared to an ordinary annuity. The formula for the future value of an annuity due incorporates this extra period by multiplying the future value of an ordinary annuity formula by (1 + r), where r is the interest rate. Therefore, to calculate the accumulated amount after ten payments of Rs. 5,000 at 16% interest, the formula FV = 5000 × {(1.16)^10 – 1} / 0.16 × (1.16) correctly accounts for this additional interest.

To determine the annual deposit needed to accumulate Rs.21,873 in 4 years at a 6% interest rate using a sinking fund, we use the formula for the future value of an annuity. The formula considers the annual deposit (PMT), the interest rate, and the number of periods. By solving for PMT, we find that depositing Rs.5,000 each year will yield the desired amount at the end of the term, accounting for the compound interest accrued over the 4 years.

The Required Rate of Return (RRR) formula accounts for the compensation an investor expects for taking on additional risk compared to a risk-free investment. It starts with the risk-free rate, which represents the return on a secure investment, and adds the market risk premium, which is the additional return expected from investing in the market over the risk-free rate. This formulation effectively captures the relationship between risk and return, making it a crucial concept in finance for evaluating investment opportunities.